The Game's Afoot, Watson!

Calculus Level 5

The multiple integral \[ \int_0^\pi \int_0^\pi \int_0^\pi \frac{du\,dv\,dw}{1 - \cos u \cos v \cos w} \] can be shown to be equal to \[ \tfrac{P}{Q} \Gamma\big(\tfrac{1}{R}\big)^S \] for positive integers \(P,Q,R,S\), where \(P,Q\) are coprime. What is the value of \(P+Q+R+S\)?

\[\] Notation: \( \Gamma(\cdot) \) denotes the Gamma function.

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